# Nonlinear embeddings for conserving Hamiltonians and other quantities with Neural Galerkin schemes

arXiv: https://arxiv.org/abs/2310.07485 · HTML: https://mlsanity.com/paper/2310.07485/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 70 references that are in the map.
- First posted on arXiv 11 Oct 2023.

## Builds on (most cited first, 16 of 16)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [An Extensible Benchmark Suite for Learning to Simulate Physical Systems](https://mlsanity.com/paper/2108.07799/) (2021)
- [Sampling low-dimensional Markovian dynamics for pre-asymptotically recovering reduced models from data with operator inference](https://mlsanity.com/paper/1908.11233/) (2019)
- [Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks](https://mlsanity.com/paper/2310.04867/) (2023)
- [Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference](https://mlsanity.com/paper/2107.02597/) (2021)
- [Canonical and Noncanonical Hamiltonian Operator Inference](https://mlsanity.com/paper/2304.06262/) (2023)
- [Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds](https://mlsanity.com/paper/2305.15490/) (2023)
- [Data-driven Structured Realization](https://mlsanity.com/paper/1611.02072/) (2016)
- [Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes](https://mlsanity.com/paper/2306.15630/) (2023)
- [Port-Hamiltonian Dynamic Mode Decomposition](https://mlsanity.com/paper/2204.13474/) (2022)
- [SOBMOR: Structured Optimization-Based Model Order Reduction](https://mlsanity.com/paper/2011.07567/) (2020)
- [Structure-preserving discretization for port-Hamiltonian descriptor systems](https://mlsanity.com/paper/1903.10451/) (2019)
- [Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems](https://mlsanity.com/paper/2303.12576/) (2023)

## Built on by (most cited first, 3 of 3)

- [CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equations](https://mlsanity.com/paper/2402.14646/) (2024)
- [A hybrid FEM-PINN method for time-dependent partial differential equations](https://mlsanity.com/paper/2409.02810/) (2024)
- [Learning Effective Dynamics across Spatio-Temporal Scales of Complex Flows](https://mlsanity.com/paper/2502.07990/) (2025)

## Abstract (arXiv)

This work focuses on the conservation of quantities such as Hamiltonians, mass, and momentum when solution fields of partial differential equations are approximated with nonlinear parametrizations such as deep networks. The proposed approach builds on Neural Galerkin schemes that are based on the Dirac--Frenkel variational principle to train nonlinear parametrizations sequentially in time. We first show that only adding constraints that aim to conserve quantities in continuous time can be insufficient because the nonlinear dependence on the parameters implies that even quantities that are linear in the solution fields become nonlinear in the parameters and thus are challenging to discretize in time. Instead, we propose Neural Galerkin schemes that compute at each time step an explicit embedding onto the manifold of nonlinearly parametrized solution fields to guarantee conservation of quantities. The embeddings can be combined with standard explicit and implicit time integration schemes. Numerical experiments demonstrate that the proposed approach conserves quantities up to machine precision.